Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C82 · Schur multiplier (H2) = C43 |
| direct product | C3×#8131 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C43 | C22 |
| commutator | C2 | C2×C3×C82 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle, prime radical | C23×C3 | C2×C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C43 ◁ G | |
574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13,14,16,15,17,18,19);(5,8)(9,11)(20,21,22,24,23,25,26,27);(4,6,7,10)(5,9,8,11)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1119);